How do I solve a differential equation with boundary conditions?

[Ted123]

Homework Statement

[PLAIN]http://img402.imageshack.us/img402/7427/diff5.jpg

The Attempt at a Solution

How do I evaluate $\frac{d}{dx} \left( \frac{y_2}{y_1} \right)$ ?

 

[Disconnected]

Homework Statement

[PLAIN]http://img402.imageshack.us/img402/7427/diff5.jpg

The Attempt at a Solution

How do I evaluate $\frac{d}{dx} \left( \frac{y_2}{y_1} \right)$ ?

Have you considered looking at the quotient rule? It seems at first glance like the numerator would be similar to W[y_1,y_2].

Then you could see that the integral of that might be a constant, and that you might be able to deduce that constant at the boundaries, and if that constant happened to be 1, then y_1 / y_2 would be 1, in which case y_1=y_2, for all x.

 

[Ted123]

So

$y_2 = \frac{y_1 y_2^'}{y_1^'}$

and

$y_1 = \frac{y_2 y_1^'}{y_2^'}$

So we want to find:

$\frac{d}{dx} \left( \frac{y_2}{y_1} \right) = \frac{d}{dx} \left( \frac{y_1 {y_2^'}^2}{y_2 {y_1^'}^2} \right)$

Letting $u = y_1 {y_2^'}^2$ and $v = y_2 {y_1^'}^2$

what is $u'$ and $v'$?

 

[Disconnected]

Calculate the derivitive directly:

[itex]\frac{d}{dx}\bigg(\frac{y_2}{y_1}\bigg)=\frac{y_2'y_1-y_2y_1'}{y_1^2}[\latex]

If we look at the numerator, it is 0. Therefore

[itex]\frac{d}{dx}\bigg(\frac{y_2}{y_1}\bigg)=0[\latex]

Therefore

[itex]\bigg(\frac{y_2}{y_1}\bigg)=C for all x[\latex]

Where C is some constant.

Look at the boundery condition at x_0,

[itex]\bigg(\frac{y_2(x_0)}{y_1(x_0)}\bigg)=1, as y_1(x_0)=y_2(x_0)[\latex]

thus C=1,

Thus

[itex]\bigg(\frac{y_2}{y_1}\bigg)=1 for all x [\latex]

Which implies that y_1=y_2 for all x!


I can't get the latex to work...
But I hope the point is clear.

 

[Ted123]

Calculate the derivitive directly:

$\frac{d}{dx}\bigg(\frac{y_2}{y_1}\bigg)=\frac{y_2' y_1-y_2y_1'}{y_1^2}$

If we look at the numerator, it is 0. Therefore

$\frac{d}{dx}\bigg(\frac{y_2}{y_1}\bigg)=0$

Therefore

$\frac{y_2}{y_1}=C=\text{const}$ for all x.

Where C is some constant.

Look at the boundery condition at x_0,

$\bigg(\frac{y_2(x_0)}{y_1(x_0)}\bigg)=1$, as $y_1(x_0)=y_2(x_0)$

thus C=1,

Thus

$\frac{y_2}{y_1} =1$ for all x

Which implies that y_1=y_2 for all x!


I can't get the latex to work...
But I hope the point is clear.

Thanks!

The latex wouldn't work as you should end it with [/itex] not [\latex] !